Question

(tan/1-cot)+(cot/1-tan)
=1+tan+cot.prove

Answer 1

To prove: $\dfrac{\tan\theta}{1-\cot\theta}+\dfrac{\cot\theta}{1-\tan\theta}=1+\tan\theta+\cot\theta$

Here we need to prove trigonometry equation.

Let me take left hand side

$\Rightarrow \dfrac{\tan\theta}{1-\cot\theta}+\dfrac{\cot\theta}{1-\tan\theta}$

$\cot\theta=\dfrac{1}{\tan\theta}$

$\Rightarrow \dfrac{\tan^2\theta}{\tan\theta-1}-\dfrac{1}{\tan\theta(\tan\theta-1)}$

$\Rightarrow \dfrac{\tan^3\theta-1}{\tan\theta(\tan\theta-1)}$

$a^3-b^3=(a-b)(a^2+ab+b^2)$

$\Rightarrow \dfrac{(\tan\theta-1)(\tan^2\theta+\tan\theta+1)}{\tan\theta(\tan\theta-1)}$

$\Rightarrow \dfrac{\tan^2\theta+\tan\theta+1}{\tan\theta}$

$\Rightarrow \tan\theta+1+\cot\theta=1+\tan\theta+\cot\theta(RHS)$

Hence Proved

Answer 2

Answer:

hey!

the answer is in picture

Answer : To prove : tan A/1-cot A + cot A/1-tan A=1+tan A+cot A

taking L.H.S

= [ tan(A) / (1 - cot(A)) ] + [ cot(A) / (1 - tan(A) ) ]

= [ sin(A)/cos(A) / (1 - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (1 - sin(A)/cos(A) ) ]

{using tan x = sinx /cos x and cotx = cosx/sinx }

= [ sin(A)/cos(A) / (sin(A)/sin(A) - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (cos(A)/cos(A) - sin(A)/cos(A) ) ]

= [ sin(A)/cos(A) / (sin(A) - cos(A)) / sin(A) ] + [ ( cos(A)/sin(A) / ( cos(A) -sin(A) ) / cos(A) ) ]

= [ sin(A)sin(A) / cos(A)(sin(A) - cos(A)) ] + [ ( cos(A)cos(A) / sin(A)( cos(A) -sin(A) ) ]

=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] + [ cos2(A) / -sin(A)( sin(A) - cos(A) ) ) ]

=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] - [ cos2(A) / sin(A)( sin(A) - cos(A) ) ) ] ===> LCD

= [ sin2(A) sin(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos2(A) cos(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]

= [ sin3(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos3(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]

=[ sin3(A) - cos3(A) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]

=[ (sin(A) - cos(A)) ( sin2(A) + sin(A) cos(A) + cos2(A) ) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]

= [ ( sin2(A) + cos2(A) + sin(A) cos(A) ] / [ sin(A)cos(A) ]

= [ sin2(A) / sin(A)cos(A) ] + [ cos2(A) / sin(A)cos(A) ] + [ sin(A) cos(A) / sin(A) cos(A) ]

= [ sin(A) / cos(A) ] + [ cos(A) / sin(A) ] + 1

= tan(A) + cot(A) + 1

{we know sin x/ cos x = tan x and cos x/sin x = cot x }

= R. H . S

Hence proved